If there is an adjunction between a category $C$ and $D$ such that the left adjoint $$L:C \to D$$ is faithful (but not full), can one describe the image of $L$ in terms of the counit of the adjunction? If it helps, in my situation the right adjoint $R$ has a further right adjoint; I am concerned with the case actually that there is faithful functor $$f:E \to F$$ and I would like characterize the image of its prolongation $$f_{!}:Set^{E^{op}} \to Set^{F^{op}}$$ (which is again faithful).
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